Existence, Uniqueness and Ergodicity of Positive Solution of Mutualism System with Stochastic Perturbation

Figure 1

The solution of system (1.1) and system (1.4) with $(x_1(0),x_2(0))=(\mathrm{1.2,3.5}),\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{11}=0.6,\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{12}=0.4,\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{21}=0.3,\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{22}=0.5,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\sigma }_1=0.02,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\sigma }_2=0.02,\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}\Delta t=0.001$ such that $a_{11}{a}_{22}>a_{12}{a}_{21}$. The imaginary lines represent the solution of system (1.1), and the real lines represent the solution of system (1.4).